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Al2O3 powders were milled in N2 atmosphere using an attritor ball mill. The phase transformation and gassolid reaction of Al2O3 during high-energy ball milling in N2 atmosphere were investigated. It is found that the -Al2O3 transform to -Al2O3 as milling time increases. Milled for 20h in N2 atmosphere, Al2O3 becomes partially amorphous and the gassolid reaction between Al2O3 and N2 takes place. A new phase with cubic aluminum nitride (AlN) structure forms during milling which is different from normal hexagonal AlN produced by carbon thermal reduction processing. As milling energy increases, the amount of this new phase with cubic AlN structure increases. The amount of this new phase is as high as 72% when Al2O3 is milled in N2 atmosphere at 650rpm for 40h.
Ball mills are among the most variable and effective tools when it comes to size reduction of hard, brittle or fibrous materials. The variety of grinding modes, usable volumes and available grinding tool materials make ball mills the perfect match for a vast range of applications.
RETSCH is the world leading manufacturer of laboratory ball mills and offers the perfect product for each application. The High Energy Ball Mill Emax and MM 500 were developed for grinding with the highest energy input. The innovative design of both, the mills and the grinding jars, allows for continuous grinding down to the nano range in the shortest amount of time - with only minor warming effects. These ball mills are also suitable for mechano chemistry. Mixer Mills grind and homogenize small sample volumes quickly and efficiently by impact and friction. These ball mills are suitable for dry, wet and cryogenic grinding as well as for cell disruption for DNA/RNA recovery. Planetary Ball Mills meet and exceed all requirements for fast and reproducible grinding to analytical fineness. They are used for the most demanding tasks in the laboratory, from routine sample processing to colloidal grinding and advanced materials development. The drum mill is a type of ball mill suitable for the fine grinding of large feed sizes and large sample volumes.
Reactive gas milling produces a homogeneous alloy microstructure.Li15Si4 formation in SiTiN alloys during cycling results in severe degradation.SiTiN with a homogeneous microstructure does not form Li15Si4 during cycling.Reactive gas milled SiTiN is an attractive alloy for Li-ion battery anodes.
SiTiN alloys were prepared by reactive gas mechanical milling of Si and Ti powders in N2(g) for use as anode materials in Li-ion cells. This is a low-cost and simple method to prepare SiTiN anode materials. As a comparison, SiTiN alloys were also synthesized more conventionally by ball milling Si and TiN powders in Ar. Both SiTiN alloys have similar phase compositions, however the SiTiN alloys prepared by N2 reactive gas milling have a more homogenous TiN distribution at the nanometer-scale. As a consequence, SiTiN alloys prepared by N2 reactive gas milling where much more resistant to Li15Si4 formation during cycling in Li-cells, resulting in improved cell performance.
When machining free-form surfaces with a ball-end milling tool, the working tool diameter is constantly changing even if the tool path is constant. The reason is that the surface normal of the milled surface is continuously changing along the milling path. When the working diameter is changing, the cutting parameters also change. This variation effects the roughness homogeneity of the smoothed surface. Simultaneous five-axis milling solves this problem; however, the price and complexity of this technology can be a problem for some industrial sectors. In the paper, the geometrical background of a solution to this problem is presented for a 3-axis ball-end milling process for machining a free form surface. The paper provides the deduction of the theory by the use of homogeneous transformations. The geometrical problem of the cutting process is treated locally where the general machined surface is substituted at every point by its tangent plane. From the result of the presented method, a milling strategy can be formulated for ball-end milling that minimises the change in the momentary working diameter thus providing a more constant cutting parameter.
The ball-end milling of free-form surfaces is one of the most often used technologies in the case of finishing. Basically, the revolving milling cutter follows the surface with a specific speed (feed rate, vf), and at the end of the surface, it steps aside and turns back. This strategy is suitable for prefinishing and finishing of non-undercut, shallow surfaces on a CNC milling machine. The strategy needs a CNC controller, and the CNC program can be generated by a CAM software based on the CAD model of the surface.
In the case of free-form surface milling with a ball-end milling cutter, the quality of surface roughness depends on several factors and parameters like the tool diameter, working diameter, cutting parameters, tool material, nature of the surface, tool path, milling direction etc.
Fan  investigates the changing of cutting speed in 3-axis ball-end milling and points out that the cutting speed is the speed factor that influences machined surface quality and tool life. de Souza et al.  show the effect of the working diameter in chip removal, cutting force and surface roughness, where higher cutting speed decreases the surface roughness. Wojciechowski et al.  analyses ball-end milling in the case of different tool inclination that results in different working diameters. In case of a large working diameter, the surface roughness is smaller, and the increasing cutting speed decreases the surface roughness quality.
Vopt et al.  present the effect of tool material and milling direction. The solid carbide end mill ensures better surface roughness and longer tool life than HSS Co, and in the case of both tool material, the down-milling results in better surface roughness. Wojciechowski et al.  present the dynamic model of ball-end milling, which can be a base of texture prognosis. Vyboishchik  presents a geometric model of surface topology in the case of a flat surface, a convex and a concave curved surface. Based on it, the inclination anglethe nature of the surfacehas an important role.
Beno et al.  analyses the main features of free-form surfaces and proposes a process to identify the most suitable milling strategy. The paper concludes that cutting speed has main effect on surface quality beside other geometric and tool parameters. Pena et al.  present 5 different tool strategies and analyses them from the point of view of surface roughness in the case of 6061 aluminium. Due to different Ra parameters, a correction factor is introduced which modifies the feed rate in order to ensure better surface roughness. de Souza et al.  present the influence of the tool path strategy on surface quality. The different strategies result in different machining time and surface quality, which has an influence on the polishing time of the surface. The different path strategies result in different machining time as it was analysed by Sales et al. . Iol et al.  present the importance of milling strategies, where it is concluded that the applied strategies have larger effect on the surface roughness than the width of cut (ae) r. Pena et al.  investigate 6 different trajectories in the case of milling inclined surfaces. They found that the calculated cusp height is smaller than the real surface deviation and suggests correction coefficient to modify the width of the cut (ae). Redonnet et al.  present an optimisation of free-form surface machining direction with a torus end mill, minimising the machining time and considering the scallop height. Zhang et al.  present a geometric model for machined surface topography in a ball-end milling process, and based on this model, optimization for 3D arithmetic average deviation (Sba) and material removal rate (MRR) was developed. The effect of feed per tooth (fz) and the radial depth of cut (ae) were considered. In addition, vibration can also effect the surface topography .
The simultaneous 5-axis milling can solve the problem of varying working diameter; however, simultaneous 5-axis milling is a very complicated process, and the machine tool is much more expensive than the 3-axis one.
In the industry, the most widely used process is the non-simultaneous 5-axis (or 3+2-axis) milling for general machining, where the machining is done in three axes, and the positioning is made with the two indexing axes, (while machining is suspended). It is a more precise and simpler process than simultaneous 5-axis machining, and the machine tools and their controllers are easier to handle and program. These processes are widely used in engine block, gear block machining, or for manufacturing non-aerodynamics part of aircrafts. This is why the three-axis machining (mainly 3+2-axis) in the mainstream industry will be an important process for a long time. This has been confirmed by serious industrial designers and manufacturers such as the development department of FFG MAG in Esslingen. Although the investigation of the paper is done for a 3-axis problem, it is still (and will be in the future) an important issue in the industry.
Xu et al.  developed a swept surface approach to model the topography of the machined surface in the case of ball-end milling. In the study, the ball-end cutter proceeds along a tool path in order to perform continuous material removal from the swept surfaces of the cutting edges. Liu et al.  present a method of surface segmentation in order to determine the best regions for a 3+2-axis milling strategy, because static rotary axis allows higher feed rate.
The surface quality according to the geometrical model depends on the tool diameter (D), the step over parameter (ae) and the inclination of the surface, also on the cutting parameters such as feed rate (vf) and cutting speed (vc). During the machining of free form surfaces, the inclination of the surface varies; therefore, the working diameter of the cutting tool is changing, thus the cutting speed also changes. The cutting speed defines the minimum value of the chip thickness. Based on our previous research ([18,19,20]), if the chip thickness is too small, because of small cutting speed, the tool cannot remove any material.
In this paper, the engineering aspects are highlighted, from the aspect of how the work of a CAM programmer can be supported. Previously, it was determined that beside other parameters, the cutting speed has an important effect on the surface roughness. In the case of ball-end milling, the working diameter can be a key element of the research.
The difference in our approach compared with the usual techniques is the pure geometric description of the working diameter. The developed mathematical model presents the milling of an inclined plane surface (with a ball-end cutter) (Fig. 1), where the tool paths are parallel to each other (Fig.2). The inclined plane is a simplified model of the locality of a free-form surface. This plane can be considered as the tangent plane of the surface at the given point.
Let the origin of the tool coordinate system be on the end-point of the ball-end cutter, and the z-axis is parallel to the rotation axis of the tool. If ap is the depth of cut and D is the diameter of the cutting tool, the effective (working) diameter is (see: Fig.3):
According to the changing of the inclination of the milled plane, this circle keeps its shape and parallel position to the plane (if the curvature of the surface is much smaller than the diameter of the tool). Nevertheless, the position and the effective diameter of the circle change related to the axis of the tool.
The contact point (Pt) and the origin of the tool coordinate system (Ot) coincide only if the milled surface is perpendicular to the axis of the tool. However, in the general case, a geometrical transformation is needed to identity position. Two rotational transformations are needed to determine the two points. The two angles of the transformation are determined relative to the surface normal vector (Fig.5), where the axes (x, y, z) refer to the workpiece frame:
Figure 6 shows a general example. The actual working diameter is the distance between the centre of the cutting tool and the contact circle. This distance depends on the direction of the milling and the direction of the scanning (feed).
Two points can be defined on the transformed circle where the angle of the tangent line in the x-y plane () is equal to the direction angle of the milling path (A). Generally, the tangent line is defined by the first differential of the curve:
The t1 and t2 parameters define the location of the intersection points on the curve, which is the possible milling diameter. There are two of these points, because these points depend on the scanning (feed) direction. (t=[0,1]). The t parameters can be determined by iteration.
During the surface milling, there is another extreme point, which is defined by the step-over parameter (ae). In the case of down-milling, the cutting process starts at point 1 and ends at point 1. In the case of up-milling, the cutting cycle starts at point 1 and ends at point 1. The positions of points 1 and 1 depend on the step-over parameter of the tool path (Fig.8).
Point 3 (Fig.9) is the intersection point of the tangent line in point 1 (e) and the perpendicular line from the point 1 to the line (e). The distance between the point 3 and point 1 should be equal to the step over distance (ae) in the x-y plane. For the step-over distance, the tangent (e) has to be translated by the distance of ae perpendicular to the milling direction. The intersections of this line with the transformed circle mark out the two extreme points (1, 1).
The first example shows the changing of the effective diameter and the cutting speed. In the presented case, the tool diameter is 10mm, the tool radius is 5mm, the depth of cut and the width of cut is 1mm, the cutting speed is 120m/min and the spindle speed is 3820 1/min. The surface orientation can be described by two angles: AN1=35, AN2=25 and the tool path orientation is A=30.
The realised cutting speed is always smaller than the designed one because the working diameter is smaller than the nominal diameter of the tool, thus 85.1m/min or 80m/min instead of 120m/min is realised. Moreover, even this speed is not constant during cutting. Depending on the cutting direction, the cutting starts either at points 1/2 or at 1/2 and finishes at the point 1/2. Although the rotation of the tool is constant, the cutting speed varies because the working diameter varies between the two points. This difference is between 20 and 42% according to the chosen parameters (Fig.11).
The second example presents the changing of the effective diameter in the function of the tool path direction (A). In Fig.13, four solutions are shown when the following parameters were applied: Dc=10mm; ap=1mm; ae=1mm; AN1=35; AN2=25.
If different tool path direction is set in the CAM system with a zigzag surface milling strategy: the working diameter and the cutting speed will be different. 1 and 1 (Fig.10) shows the working diameter in the case of forward and backward motion, when the milling is performed from bottom to top (Z+ direction). The 2 and 2 (Fig.10) shows the values of the diameter in the case of top to down milling (Z-direction).
In the case of the investigated surface inclination, the smallest differences between the two diameters are A=90 (1-1) and A=0 (2-2). The value of the difference and the position of the minimum value depends on the surface inclination (AN1, AN2) and also on the cutting parameters (ap, ae).
The third example shows the effect of the changing of the surface orientation to the effective diameter when the following parameters were applied Dc=10mm; ap=1mm; ae=1mm; the tool path direction is A=60, the second component of the surface normal vector is changed (AN1=30; AN2=45045).
If the inclination of the surface is changed (as in free-form surfaces), the working diameter also changes; they are equal only in the case if the plane is perpendicular to the Z axis. The bottom to top milling is more favourable, because the difference between the two diameters is smaller (Fig.14).
According to the presented method, it can be stated that the working diameter of a ball-end milling tool can be calculated by taking into consideration the parameters of the cutting path (direction and density) and the characteristics of the surface (surface normals). The results of the calculations provide four solutions. By considering the milling direction and the steps of the milling process, only one solution is realised on the course of a given milling path. Along the milling path, provided the direction of the feed does not change (A) meaning that the tool travels in a straight line, the working diameter changes because the surface normal (AN1, AN2) changes. It is clear from the results of the sample calculations that the changes are continuous and periodical, but for practical reasons, it is unnecessary to consider the full range. For practical values of each parameter for the examined angles are:
Angle A can be any angle between 0 and 360; however, all the solutions considering angles larger than 180 is already given in the original solution. These solutions represent the same milling in an opposite direction.
The working diameter of a ball-end milling tool is continuously changing during the milling of a free-form surface with a ball-end cutter; thus the actual position of the cutting point on the tool edge is also changing. Together with this parameter, the chip cross section, the cutting force and the effective cutting speed also change. These circumstances and parameters have effect on the micro- and macro precision of the surface too. To minimalize this effect, it is worth considering keeping the working diameter constant or at a very slightly varying value during the course of tool path planning. A solution for this problem can be the simultaneous 5-axis milling technology. However, this technology is not always applicable or justified for technical and economic reasons; this is why the tree-axis milling will still be a possible solution for a long time to come.
By the proper planning of the milling path, the negative effect of the changing of the working diameter can be considerably reduced. For this, it is necessary to calculate the working diameter at any point of the path on the surface. This is shown in Fig.15, where this feedback is illustrated with dashed lines.
Since the design of the toolpath has its constraints and limitations, using only this method will not provide a full solution. In the case of rapidly changing surfaces, the speed variations could be too large thus difficult to apply. It could be a good solution of combining the two strategies in order to create a more homogeneous surface for roughness.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Mik, B., Zentay, P. A geometric approach of working tool diameter in 3-axis ball-end milling. Int J Adv Manuf Technol 104, 14971507 (2019). https://doi.org/10.1007/s00170-019-03968-9